When I zone into Drustvar, it tells me to loot the 4 cursed chests for the pet parts. But this character has the achievement, and the pet is in my journal.

I believe the Scouting Reports have been fixed by being tracked as a 'once-per-account' quest, so they should hide once completed on any character.
I'm about to push a fix for the above treasure for the pet by using our newer tech in the addon to properly track it.

also all of the "Scouting Reports"

Can you screenshot what specifically seems to be the error? What is your current Tracking setup in the first tab of the Settings? It's likely that you actually completed the pet on a different character and are not tracking Quests with 'Account-Wide', causing the treasures/items to show for other characters.

The Scouting Reports unfortunately do not have a decent solution at this time, as Blizzard made it so that they unlock Account-Wide when establishing the respective Outposts, but not giving Quest completion to any other character other than the one which actually established the Outpost. No one has been able to properly complete any stage of those Quests via Party Sync or other means once their Account has established an Outpost, so the likely solution will require ATT to 'pretend' that a character has completed all of the Quests by checking for some other criteria to determine that it's impossible to acquire and complete those Quests. But again, that criteria for this check has also not yet been determined and with Shadowlands taking priority presently, I don't see that happening for some time...

These are marked as open, but the secondary items not collected. Achievement is marked off in game and completed with this character. Still marked incomplete if I toggle account wide quests.
It appears it was checking for the uncommon version of the pet as mine had been upgraded to rare. I went and got another on a different character and it now disappears when I toggle account wide quest tracking on. Id prefer not to use that personally, but it does work as a workaround.

As for the Scouting Reports, im just happy its a known thing and will be worked on eventually! :)